# M3: Introductory Calculus - Material for the year 2019-2020

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16 lectures

These lectures are designed to give students a gentle introduction to applied mathematics in their first term at Oxford, allowing time for both students and tutors to work on developing and polishing the skills necessary for the course. It will have an `A-level' feel to it, helping in the transition from school to university. The emphasis will be on developing skills and familiarity with ideas using straightforward examples.

At the end of the course, students will be able to solve a range of ordinary differential equations (ODEs). They will also be able to evaluate partial derivatives and use them in a variety of applications.

See the examinable syllabus.

General linear homogeneous ODEs: integrating factor for first order linear ODEs, second solution when one solution is known for second order linear ODEs. First and second order linear ODEs with constant coefficients. General solution of linear inhomogeneous ODE as particular solution plus solution of homogeneous equation. Simple examples of finding particular integrals by guesswork. [4]

Introduction to partial derivatives. Second order derivatives and statement of condition for equality of mixed partial derivatives. Chain rule, change of variable, including planar polar coordinates. Solving some simple partial differential equations (e.g. $f_{xy} = 0$, $f_x = f_y$). [3.5]

Parametric representation of curves, tangents. Arc length. Line integrals. [1]

Jacobians with examples including plane polar coordinates. Some simple double integrals calculating area and also $\int_{\mathbb{R}^2} e^{-(x^2+y^2)} dA$. [2]

Simple examples of surfaces, especially as level sets. Gradient vector; normal to surface; directional derivative; $\int^B_A \nabla \phi \cdot d\mathbf{r} = \phi(B)-\phi(A)$.[2]

Taylor's Theorem for a function of two variables (statement only). Critical points and classification using directional derivatives and Taylor's theorem. Informal (geometrical) treatment of Lagrange multipliers.[3.5]

1) M. L. Boas, *Mathematical Methods in the Physical Sciences* (Wiley, 3rd Edition, 2005).

2) D. W. Jordan & P. Smith, *Mathematical Techniques* (Oxford University Press, 3rd Edition, 2003).

3) E. Kreyszig, *Advanced Engineering Mathematics* (Wiley, 10th Edition, 2011).

4) K. A. Stroud, *Advanced Engineering Mathematics* (Palgrave Macmillan, 5th Edition, 2011).